This paper addresses a numerical method for predicting transient temperature distributions in the wall of a curved pipe subjected to internal laminar thermally-stratified flow. A simple and convenient numerical method of treating the unsteady conjugate heat transfer in non-orthogonal coordinate systems is presented. Numerical calculations are performed for the transient evolution of thermal stratification in two curved pipes, where one has thick wall and the other has so thin wall that its presence can be negligible in the heat transfer analysis. The predicted results show that the thermally stratified flow and transient conjugate heat transfer in a curved pipe with a finite wall thickness can be satisfactorily analyzed by the present numerical method, and that the neglect of wall thickness in the prediction of pipe wall temperature distributions can provide unacceptably distorted results for the cases of pipes with thick wall such as safety related-piping systems of nuclear power plant.